二维可积场论的计算同调方法
Computational homological methods for integrable field theories
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中文总结 AI 辅助
研究从四维半全纯陈 - 西蒙斯理论构建二维可积场论的同调方法，通过循环\(L_\infty\)代数同伦转移积分谱曲线，为\(C = \mathbb{C}P^1\)构造强形变收缩，计算主手征模型相关量，恢复标准作用与拉克斯联络。
中文摘要 AI 辅助
我们为从\(\Sigma\times C\)上的四维半全纯陈 - 西蒙斯理论构建\(\Sigma\)上的二维可积场论的近期同调方法开发了显式计算工具。在此框架下，通过与具有规定奇点和边界条件的四维理论相关的循环\(L_\infty\)代数的同伦转移来实现对谱曲线\(C\)的积分操作。我们为\(C = \mathbb{C}P^1\)上的除子扭曲多贝尔复形构造了显式强形变收缩，并利用它们使转移后的\(L_\infty\)结构在计算上可及。作为应用，我们研究了对应于具有韦斯 - 祖米诺项的主手征模型的亚纯\(1\)-形式的选择。我们计算了转移后的毛雷尔 - 卡丹作用和相关的拉克斯联络，表明前者恢复为具有韦斯 - 祖米诺项的标准主手征模型作用，后者再现了通常的拉克斯联络。
英文摘要
We develop explicit computational tools for the recent homological approach to the construction of $2$-dimensional integrable field theories on $Σ$ from $4$-dimensional semi-holomorphic Chern-Simons theory on $Σ\times C$. In this framework, the operation of integrating out the spectral curve $C$ is realized by homotopy transfer of a cyclic $L_\infty$-algebra associated with the $4$-dimensional theory with prescribed singularities and boundary conditions. We construct explicit strong deformation retracts for divisor-twisted Dolbeault complexes on $C=\mathbb{C}P^1$ and use them to make the transferred $L_\infty$-structure computationally accessible. As an application, we study the choice of meromorphic $1$-form corresponding to the principal chiral model with a Wess-Zumino term. We compute the transferred Maurer-Cartan action and the associated Lax connection, showing that the former resums to the standard principal chiral model action with a Wess-Zumino term and that the latter reproduces the usual Lax connection.
